The longer time is spent by the departure of the U.K. from the EU, the more this debate about the departure will be on the political radar of everybody which naturally makes the advocates of other "exits" more visible and important.
If the U.K. is fully removed from the EU soon, the EU officials may try to quickly restore the old "business-as-usual" pretending that nothing serious has happened, no changes in the EU are necessary, the U.K. has de facto never belonged to the EU so it's not a big deal, and the EU should continue along the pre-Brexit path.
It's similar to a boyfriend who is rejected by a girlfriend and wants to stay active. He may want to get rid of the memories as soon as possible because they may prevent him from finding a new girlfriend etc.
This is probably how the EU officials are really thinking (Merkel seems to be an exception, doesn't insist on speed) but it's highly disputable whether this strategy is realistic. There is a widespread opinion that the EU simply cannot continue along the same path towards ever deeper integration and ever greater arrogance of power that we observed before the referendum.
Everyone has realized how strong the voters in many countries actually are and that the Euroskepticism is very real. It is reasonable to expect that this newly discovered power of the Euroskeptics won't be forgotten. The Brits have created a precedent that won't go away. So in practice, it doesn't really matter how quickly the U.K. leaves the union.
In practice, the departure could be and should be much faster than David Cameron wants to suggest. The negotiations about the dissolution of Czechoslovakia only began after elections in mid 1992 and everything was absolutely ready by the end of the year 1992 – in less than 6 months – and new countries were fully created on January 1st, 1993. The exodus of cash from Slovakia caused concerns so the currencies had to be divided as well, just 6 weeks later, and it basically took just days of instant planning.
Czechoslovakia was a single country – the U.K. is just "vaguely affiliated" through the EU. So the U.K. departure could be expected to be simpler, not harder, than the Czechoslovak Velvet Divorce.